Is it permissible to pay a sum of money to an employee at the Authority to change my job classification to match the job title at the hospital, in order to enable me to obtain the deserved promotions, especially since the date I obtained my certificate precedes the Authority's new decisions?

Bribery is the payment of money to achieve something false or unjust. However, paying money to attain a right or avert an injustice is not forbidden bribery for the giver, and the sin falls upon the receiver. If you are entitled to have the classification changed, then there is no harm in paying the amount; otherwise, it is not permissible.